3.7 Definite integrals

(a) $-f(x)$ (b) $f(x+1) - f(x)$ (c) $\displaystyle{2x \int_{0}^{2x}f(t)dt + 2x^2 f(2x)}$

(a) $\displaystyle{\frac{32}{3}}$ (b) $\displaystyle{\frac{1}{4}}$ (c) $1$ (d) $\displaystyle{\frac{1}{2} - \frac{1}{2 e}}$ (e) $\displaystyle{\log{(\frac{3}{2})}}$

3. omitted

(a) $\displaystyle{\frac{\pi}{4} = \int_{0}^{1} \frac{dx}{1+x^2} \leq \int_{0}^{1} \frac{dx}{1 + x^{n}} < 1}$

(b)

$\displaystyle \frac{1}{2(n+1)} < \int_{0}^{1}\frac{x^{n}}{1+x} dx < \frac{1}{n+1} < \frac{1}{n}$

(a) $\displaystyle{\log{2}}$ (b) $\displaystyle{\log{(1 + \sqrt{2})}}$ (c) $\displaystyle{\frac{1}{3}}$